Relational Magnitudes

Relational Magnitudes are invariant vectorial quantities that conserve their value and form under transformations of translation and rotation.

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I. Definitions I (Relational Magnitudes)

The relational position ($\mathbf{r}_i$), relational velocity ($\mathbf{v}_i$), and relational acceleration ($\mathbf{a}_i$) of a particle $i$ with respect to an Auxiliary Reference Frame, are given by:

$\mathbf{r}_i \doteq \vec{r}_i$

$\mathbf{v}_i \doteq \dfrac{d(\vec{r}_i)}{dt} = \vec{v}_i$

$\mathbf{a}_i \doteq \dfrac{d^2(\vec{r}_i)}{dt^2} = \vec{a}_i$

Where $\vec{r}_i$, $\vec{v}_i$ and $\vec{a}_i$ are the ordinary vectorial position, velocity, and acceleration of the particle $i$ with respect to the Auxiliary Reference Frame.

Note

The Relational (Vectorial) Magnitudes are always the same as the Ordinary (Vectorial) Magnitudes in the Auxiliary Reference Frame.

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II. Definitions II (Relational Magnitudes)

The relational position ($\mathbf{r}_i$), relational velocity ($\mathbf{v}_i$), and relational acceleration ($\mathbf{a}_i$) of a particle $i$ with respect to any Reference Frame $S$, are given by:

$\mathbf{r}_i \doteq \vec{r}_i - \vec{R}$

$\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R})$

$\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R})$

Where:

• $\vec{r}_i, \vec{v}_i, \vec{a}_i$ are the ordinary vectorial position, velocity, and acceleration of particle $i$ with respect to the Frame $S$.
• $\vec{R}, \vec{V}, \vec{A}$ are the position, velocity, and acceleration of the Auxiliary Frame's origin with respect to $S$.
• $\vec{\omega}$ and $\vec{\alpha}$ are the angular velocity and angular acceleration of the Auxiliary Frame with respect to $S$.

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III. Transformations (Invarianza$\cdot$Relations)

The transformations of relational position, relational velocity and relational acceleration of a particle $i$ between a Reference Frame $S$ and another Reference Frame $S'$, are given by:

$\mathbf{r}_i \doteq (\vec{r}_i - \vec{R}) = \mathbf{r}'_i$

$\mathbf{r}'_i \doteq (\vec{r}'_i - \vec{R}') = \mathbf{r}_i$

$\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R}) = \mathbf{v}'_i$

$\mathbf{v}'_i \doteq (\vec{v}'_i - \vec{V}') - \vec{\omega}' \times (\vec{r}'_i - \vec{R}') = \mathbf{v}_i$

$\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R}) = \mathbf{a}'_i$

$\mathbf{a}'_i \doteq (\vec{a}'_i - \vec{A}') - 2\vec{\omega}' \times (\vec{v}'_i - \vec{V}') + \vec{\omega}' \times [\ \vec{\omega}' \times (\vec{r}'_i - \vec{R}')\ ] - \vec{\alpha}' \times (\vec{r}'_i - \vec{R}') = \mathbf{a}_i$

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IV. Bibliography

1. A. Blatter, A Reformulation of Classical Mechanics (2015). PDF
2. A. Tobla, A Reformulation of Classical Mechanics (2024). PDF